Integrand size = 28, antiderivative size = 260 \[ \int \frac {(d \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=-\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}-\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}+\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {d \sqrt {d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))} \]
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Time = 0.33 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3631, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {(d \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=-\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) d^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} a f}-\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) d^{3/2} \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a f}+\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) d^{3/2} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{\sqrt {2} a f}-\frac {d \sqrt {d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))} \]
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Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3615
Rule 3631
Rubi steps \begin{align*} \text {integral}& = -\frac {d \sqrt {d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))}+\frac {\int \frac {\frac {a d^2}{2}-\frac {3}{2} i a d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{2 a^2} \\ & = -\frac {d \sqrt {d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))}+\frac {\text {Subst}\left (\int \frac {\frac {a d^3}{2}-\frac {3}{2} i a d^2 x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 f} \\ & = -\frac {d \sqrt {d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))}+\frac {\left (\left (\frac {1}{4}-\frac {3 i}{4}\right ) d^2\right ) \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f}+\frac {\left (\left (\frac {1}{4}+\frac {3 i}{4}\right ) d^2\right ) \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f} \\ & = -\frac {d \sqrt {d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))}+-\frac {\left (\left (\frac {1}{8}+\frac {3 i}{8}\right ) d^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}+-\frac {\left (\left (\frac {1}{8}+\frac {3 i}{8}\right ) d^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}+\frac {\left (\left (\frac {1}{8}-\frac {3 i}{8}\right ) d^2\right ) \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f}+\frac {\left (\left (\frac {1}{8}-\frac {3 i}{8}\right ) d^2\right ) \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a f} \\ & = -\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}+\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {d \sqrt {d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))}+-\frac {\left (\left (\frac {1}{4}-\frac {3 i}{4}\right ) d^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\left (\frac {1}{4}-\frac {3 i}{4}\right ) d^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f} \\ & = -\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}+\frac {\left (\frac {1}{4}-\frac {3 i}{4}\right ) d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} a f}-\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}+\frac {\left (\frac {1}{8}+\frac {3 i}{8}\right ) d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} a f}-\frac {d \sqrt {d \tan (e+f x)}}{2 f (a+i a \tan (e+f x))} \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.43 \[ \int \frac {(d \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=\frac {d \left (\sqrt [4]{-1} \sqrt {d} \arctan \left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )-2 \sqrt [4]{-1} \sqrt {d} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )+\frac {i \sqrt {d \tan (e+f x)}}{-i+\tan (e+f x)}\right )}{2 a f} \]
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Time = 0.73 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.36
method | result | size |
derivativedivides | \(\frac {2 d^{2} \left (-\frac {\sqrt {d \tan \left (f x +e \right )}}{4 \left (i d \tan \left (f x +e \right )+d \right )}-\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{2 \sqrt {-i d}}-\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{4 \sqrt {i d}}\right )}{f a}\) | \(93\) |
default | \(\frac {2 d^{2} \left (-\frac {\sqrt {d \tan \left (f x +e \right )}}{4 \left (i d \tan \left (f x +e \right )+d \right )}-\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {-i d}}\right )}{2 \sqrt {-i d}}-\frac {i \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {i d}}\right )}{4 \sqrt {i d}}\right )}{f a}\) | \(93\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (186) = 372\).
Time = 0.25 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.02 \[ \int \frac {(d \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=\frac {{\left (a f \sqrt {-\frac {i \, d^{3}}{4 \, a^{2} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {2 \, {\left (i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {i \, d^{3}}{4 \, a^{2} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{d}\right ) - a f \sqrt {-\frac {i \, d^{3}}{4 \, a^{2} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {2 \, {\left (i \, d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 2 \, {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {i \, d^{3}}{4 \, a^{2} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{d}\right ) + a f \sqrt {\frac {i \, d^{3}}{a^{2} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (\frac {{\left (i \, d^{2} + {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {i \, d^{3}}{a^{2} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a f}\right ) - a f \sqrt {\frac {i \, d^{3}}{a^{2} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (\frac {{\left (i \, d^{2} - {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {i \, d^{3}}{a^{2} f^{2}}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{a f}\right ) - {\left (d e^{\left (2 i \, f x + 2 i \, e\right )} + d\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a f} \]
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\[ \int \frac {(d \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=- \frac {i \int \frac {\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\tan {\left (e + f x \right )} - i}\, dx}{a} \]
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Exception generated. \[ \int \frac {(d \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.48 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.67 \[ \int \frac {(d \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=-\frac {1}{2} \, d^{2} {\left (\frac {i \, \sqrt {2} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a \sqrt {d} f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {2 i \, \sqrt {2} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{-4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{a \sqrt {d} f {\left (-\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} - \frac {i \, \sqrt {d \tan \left (f x + e\right )}}{{\left (d \tan \left (f x + e\right ) - i \, d\right )} a f}\right )} \]
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Time = 6.83 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.53 \[ \int \frac {(d \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx=-\mathrm {atan}\left (\frac {2\,a\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {\frac {d^3\,1{}\mathrm {i}}{4\,a^2\,f^2}}}{d^2}\right )\,\sqrt {\frac {d^3\,1{}\mathrm {i}}{4\,a^2\,f^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {4\,a\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-\frac {d^3\,1{}\mathrm {i}}{16\,a^2\,f^2}}}{d^2}\right )\,\sqrt {-\frac {d^3\,1{}\mathrm {i}}{16\,a^2\,f^2}}\,2{}\mathrm {i}-\frac {d^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}{2\,a\,f\,\left (-d\,\mathrm {tan}\left (e+f\,x\right )+d\,1{}\mathrm {i}\right )} \]
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